MECHANICS Kinematics Chapter 2 Motion along a straight line I Position and displacement II Velocity III Acceleration IV Motion in one dimension with constant acceleration V Free fall Particle point like object that has a mass but infinitesimal size I Position and displacement Position Defined in terms of a frame of reference x or y axis in 1D The object s position is its location with respect to the frame of reference Position Time graph shows the motion of the particle car The smooth curve is a guess as to what happened between the data points 1 I Position and displacement Displacement Change from position x1 to x2 during a time interval x x2 x1 2 1 Vector quantity Magnitude absolute value and direction sign Coordinate position Displacement x x x x Coordinate system x1 x2 x2 x1 t x 0 t x 0 Only the initial and final coordinates influence the displacement many different motions between x1and x2 give the same displacement Distance length of a path followed by a particle Scalar quantity Displacement Distance Example round trip house work house distance traveled 10 km displacement 0 Review Vector quantities need both magnitude size or numerical value and direction to completely describe them We will use and signs to indicate vector directions Scalar quantities are completely described by magnitude only 2 II Velocity Average velocity Ratio of the displacement x that occurs during a particular time interval t to that interval v avg x x 2 x1 t t 2 t1 Motion along x axis 2 2 Vector quantity indicates not just how fast an object is moving but also in which direction it is moving SI Units m s Dimensions Length Time L T The slope of a straight line connecting 2 points on an x versus t plot is equal to the average velocity during that time interval Average speed Total distance covered in a time interval Savg Total distance t 2 3 Savg magnitude Vavg Savg always 0 Scalar quantity Same units as velocity Example A person drives 4 mi at 30 mi h and 4 mi and 50 mi h Is the average speed 40 mi h 40 mi h t1 4 mi 30 mi h 0 13 h t2 4 mi 50 mi h 0 08 h ttot 0 213 h Savg 8 mi 0 213h 37 5mi h 3 Instantaneous velocity How fast a particle is moving at a given instant vx lim t 0 x dx t dt 2 4 Vector quantity The limit of the average velocity as the time interval becomes infinitesimally short or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time Can be positive negative or zero x t The instantaneous velocity is the slope of the line tangent to the x vs t curve green line t Instantaneous velocity Position Slope of the particle s position time curve at a given instant of time V is tangent to x t when t 0 Time When the velocity is constant the average velocity over any time interval is equal to the instantaneous velocity at any time Instantaneous speed Magnitude of the instantaneous velocity Example car speedometer Scalar quantity Average velocity or average acceleration always refers to an specific time interval Instantaneous velocity acceleration refers to an specific instant of time 4 III Acceleration Average acceleration Ratio of a change in velocity v to the time interval t in which the change occurs aavg v2 v1 v t 2 t1 t 2 5 Vector quantity V t Dimensions L T 2 Units m s2 The average acceleration in a v t plot is the slope of a straight line connecting points corresponding to two different times t t Instantaneous acceleration Limit of the average acceleration as t approaches zero Vector quantity v dv d 2 x t 0 t dt dt 2 a lim 2 6 The instantaneous acceleration is the slope of the tangent line v t plot at a particular time green line in B Average acceleration blue line When an object s velocity and acceleration are in the same direction same sign the object is speeding up When an object s velocity and acceleration are in the opposite direction the object is slowing down 5 Positive acceleration does not necessarily imply speeding up and negative acceleration slowing down Example 1 v1 25m s v2 0m s in 5s particle slows down aavg 5m s2 An object can have simultaneously v 0 and a 0 Example 2 x t At2 v t 2At a t 2A At t 0s v 0 0 but a 0 2A Example 3 The car is moving with constant positive velocity red arrows maintaining same size Acceleration equals zero Example 4 acceleration velocity Velocity and acceleration are in the same direction a is uniform blue arrows of same length Velocity is increasing red arrows are getting longer Example 5 acceleration velocity Acceleration and velocity are in opposite directions Acceleration is uniform blue arrows same length Velocity is decreasing red arrows are getting shorter 6 IV Motion in one dimension with constant acceleration Average acceleration and instantaneous acceleration are equal a aavg v v0 t 0 Equations for motion with constant acceleration v v 0 at v avg v avg 2 7 t x x0 x x 0 v avg t 2 8 t v v at 0 and 2 7 v avg v 0 2 2 2 8 2 9 x a xa0 vv 0 tv 0 avg t 0 at 2 2 9 2 2 10 2 7 2 10 v 2 v 02 a 2 t 2 2 a v 0 t v 02 a 2 t 2 2 a x x 0 v 2 v 02 2 a x x 0 at 2 2 t 2 11 t missing t PROBLEMS Chapter 2 P1 A red car and a green car move toward each other in adjacent lanes and parallel to The x axis At time t 0 the red car is at x 0 and the green car at x 220 m If the red car has a constant velocity of 20km h the cars pass each other at x 44 5 m and if it has a constant velocity of 40 km h they pass each other at x 76 6m What are a the initial velocity and b the acceleration of the green car vr2 40km h vr1 20km h 3 km 1h 10 m 11 11m s 40 h 3600s 1km Xr2 76 6m x O Xr1 44 5 m d 220 m 44 5m x v t t 8s r1 r1 1 1 5 55m s 76 6m x v t t2 6 9s r2 r2 2 11 11m s Xg 220m x x v t r r0 r 1 x x v t at 2 g g0 g0 2 1 2 x x g v t 0 5 a g t2 2 76 6 220 v 6 9s 0 5 6 9s 2 a g r2 g0 2 g0 x x g v t 0 5 ag t12 44 5 220 v 8s 0 …